We consider mathematical modeling of problems with free boundaries, namely, the investigation of the Stefan type problem with a non-constant melting point, taking into account the pressure in the liquid phase. A mathematical model has been constructed that describes the melting process of heavy paraffin-base crude oil. We prove an existence and uniqueness of a solution of a Stefan-type problem in the Holder spaces with a melting point that depends on the unknown pressure are considered. We study the solvability in Holder spaces and the asymptotic behavior for t, which tends to $\mathrm{\infty }$, solving a single-phase Stefan-type problem for a nonlinear parabolic equation.

中文翻译：

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液相压力未知的Stefan问题的研究

我们考虑具有自由边界的问题的数学建模，即考虑到液相压力，研究具有非恒定熔点的Stefan型问题。已经建立了描述重链烷烃基原油熔融过程的数学模型。我们证明了在Holder空间中考虑熔点取决于未知压力的Stefan型问题的解的存在性和唯一性。我们研究了Holder空间中的可解性和t的渐近行为，这倾向于$\mathrm{\infty }$，求解非线性抛物方程的单相Stefan型问题。